On compact bicrossed products

We study compact bicrossed product arising from a matched pair of a discrete group Γ and a compact group G. We exhibit an automatic regularity property of the matched pair and describe the representation theory and the fusion rules of such bicrossed product. We characterize the relative Kazhdan property (T) and the relative Haagerup property of the pair given by G and the bicrossed product in terms of the action of Γ on G. We also provide an explicit example of a non-trivial discrete quantum group with the Kazhdan property (T). Finally, we study all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also estbalish the permanence of rapid decay and weak amenability and some several explicit examples